Low-dimensional Control Research Articles

Disturbance Rejection Fuzzy Control for Nonlinear Parabolic PDE Systems via Multiple Observers

A design method of low-dimensional disturbance rejection fuzzy control (DRFC) via multiple observers is proposed for a class of nonlinear parabolic partial differential equation (PDE) systems, where the disturbance is modeled by an exosystem of ordinary differential equations (ODEs) and enters into the PDE system through the control channel. In the proposed scheme, the modal decomposition technique is initially applied to the PDE system to derive a slow subsystem of low-dimensional nonlinear ODEs, which accurately captures the dominant dynamics of the PDE system. The resulting nonlinear slow subsystem is subsequently represented by a Takagi–Sugeno (T–S) fuzzy model. From the T–S fuzzy model and the exosystem, a fuzzy slow mode observer and a fuzzy disturbance observer are constructed to estimate the slow mode and the disturbance, respectively. Furthermore, a nonlinear observation spillover observer is proposed to compensate the effect of observation spillover. Then, based on these observers, a low-dimensional DRFC design is developed in terms of linear matrix inequalities to guarantee the exponential stability of the closed-loop PDE system in the presence of the disturbance. Finally, the effectiveness of the proposed design method is demonstrated on the control of one-dimensional Burgers–KPP–Fisher diffusion–reaction system and the temperature profile of a catalytic rod.

An adaptive low-dimensional control to compensate for actuator redundancy and FES-induced muscle fatigue in a hybrid neuroprosthesis

To restore walking and standing function in persons with paraplegia, a hybrid walking neuroprosthesis that combines a powered exoskeleton and functional electrical stimulation (FES) can be more advantageous than sole FES or powered exoskeleton technologies. However, the hybrid actuation structure introduces certain control challenges: actuator redundancy, cascaded muscle activation dynamics, FES-induced muscle fatigue, and unmeasurable states. In this paper, a human motor control inspired control scheme is combined with a dynamic surface control method to overcome these challenges. The new controller has an adaptive muscle synergy-based feedforward component which requires a fewer number of control signals to actuate multiple effectors in a hybrid neuroprosthesis. In addition, the feedforward component has an inverse fatigue signal to counteract the effects of the muscle fatigue. A dynamic surface control (DSC) method is used to deal with the cascaded actuation dynamics without the need for acceleration signals. The DSC structure was modified with a delay compensation term to deal with the electromechanical delays due to FES. A model based estimator is used to estimate the unmeasurable fatigue and actuator activation signals. The development of the controller and a Lyapunov stability analysis, which yielded semi-global uniformly ultimately boundedness, are presented in the paper. Computer simulations were performed to test the new controller on a 2 degrees of freedom fixed hip model after which preliminary experiments were conducted on one able-bodied male subject in the fixed hip configuration.

Active vibration control of thin-plate structures with partial SCLD treatment

To effectively suppress the low-frequency vibration of a thin-plate, the strategy adopted is to develop a model-based approach to the investigation on the active vibration control of a clamped–clamped plate with partial SCLD treatment. Firstly, a finite element model is developed based on the constitutive equations of elastic, piezoelectric and viscoelastic materials. The characteristics of viscoelastic materials varying with temperature and frequency are described by GHM damping model. A low-dimensional real modal control model which can be used as the basis for active vibration control is then obtained from the combined reduction. The emphasis is placed on the feedback control system to attenuate the vibration of plates with SCLD treatments. A modal controller in conjunction with modal state estimator is designed to solve the problem of full state feedback, making it much more feasible to real-time control. Finally, the theoretical model is verified by modal test, and an active vibration control is validated by hardware-in-the-loop experiment under different external excitations. The numerical and experimental study demonstrate how the piezoelectric actuators actively control the lower modes (first bending and torsional modes) using modal controller, while the higher frequency vibration attenuated by viscoelastic passive damping layer.

A Muscle Synergy-Inspired Adaptive Control Scheme for a Hybrid Walking Neuroprosthesis.

A hybrid neuroprosthesis that uses an electric motor-based wearable exoskeleton and functional electrical stimulation (FES) has a promising potential to restore walking in persons with paraplegia. A hybrid actuation structure introduces effector redundancy, making its automatic control a challenging task because multiple muscles and additional electric motor need to be coordinated. Inspired by the muscle synergy principle, we designed a low dimensional controller to control multiple effectors: FES of multiple muscles and electric motors. The resulting control system may be less complex and easier to control. To obtain the muscle synergy-inspired low dimensional control, a subject-specific gait model was optimized to compute optimal control signals for the multiple effectors. The optimal control signals were then dimensionally reduced by using principal component analysis to extract synergies. Then, an adaptive feedforward controller with an update law for the synergy activation was designed. In addition, feedback control was used to provide stability and robustness to the control design. The adaptive-feedforward and feedback control structure makes the low dimensional controller more robust to disturbances and variations in the model parameters and may help to compensate for other time-varying phenomena (e.g., muscle fatigue). This is proven by using a Lyapunov stability analysis, which yielded semi-global uniformly ultimately bounded tracking. Computer simulations were performed to test the new controller on a 4-degree of freedom gait model.

A model-based approach to predict muscle synergies using optimization: application to feedback control.

This paper presents a new model-based method to define muscle synergies. Unlike the conventional factorization approach, which extracts synergies from electromyographic data, the proposed method employs a biomechanical model and formally defines the synergies as the solution of an optimal control problem. As a result, the number of required synergies is directly related to the dimensions of the operational space. The estimated synergies are posture-dependent, which correlate well with the results of standard factorization methods. Two examples are used to showcase this method: a two-dimensional forearm model, and a three-dimensional driver arm model. It has been shown here that the synergies need to be task-specific (i.e., they are defined for the specific operational spaces: the elbow angle and the steering wheel angle in the two systems). This functional definition of synergies results in a low-dimensional control space, in which every force in the operational space is accurately created by a unique combination of synergies. As such, there is no need for extra criteria (e.g., minimizing effort) in the process of motion control. This approach is motivated by the need for fast and bio-plausible feedback control of musculoskeletal systems, and can have important implications in engineering, motor control, and biomechanics.

Extracting motor synergies from random movements for low-dimensional task-space control of musculoskeletal robots

In the field of human motor control, the motor synergy hypothesis explains how humans simplify body control dimensionality by coordinating groups of muscles, called motor synergies, instead of controlling muscles independently. In most applications of motor synergies to low-dimensional control in robotics, motor synergies are extracted from given optimal control signals. In this paper, we address the problems of how to extract motor synergies without optimal data given, and how to apply motor synergies to achieve low-dimensional task-space tracking control of a human-like robotic arm actuated by redundant muscles, without prior knowledge of the robot. We propose to extract motor synergies from a subset of randomly generated reaching-like movement data. The essence is to first approximate the corresponding optimal control signals, using estimations of the robot's forward dynamics, and to extract the motor synergies subsequently. In order to avoid modeling difficulties, a learning-based control approach is adopted such that control is accomplished via estimations of the robot's inverse dynamics. We present a kernel-based regression formulation to estimate the forward and the inverse dynamics, and a sliding controller in order to cope with estimation error. Numerical evaluations show that the proposed method enables extraction of motor synergies for low-dimensional task-space control.

Representation of Muscle Synergies in the Primate Brain.

We studied the motor cortical and forearm muscular activity of rhesus macaques (Macaca mulatta) as they reached, grasped, and carried objects of varied shape and size. We applied non-negative matrix factorization separately to the cortical and muscular data to reduce their dimensionality to a smaller set of time-varying "spatiotemporal" synergies. Each synergy represents a sequence of cortical or muscular activity that recurred frequently during the animals' behavior. Salient features of the synergies (including their dimensionality, timing, and amplitude modulation) were observed at both the cortical and muscular levels. The possibility that the brain may execute even complex behaviors using spatiotemporal synergies has implications for neuroprosthetic algorithm design, which could become more computationally efficient by adopting the discrete and low-dimensional control that they afford.

A body-machine interface for training selective pelvis movements in stroke survivors: A pilot study.

The body-machine interfaces (BMIs) map the subjects' movements into the low dimensional control space of external devices to reach assistive and/or rehabilitative goals. This work is a first proof of concept of this kind of BMI as tool for rehabilitation after stroke. We designed an exercise to improve the control of selective movements of the pelvis in stroke survivors, increasing the ability to decouple the motion in the sagittal and frontal planes and decreasing compensatory adjustments at the shoulder girdle. A Kinect sensor recorded the movements of the subjects. Subjects played different games by controlling the vertical and horizontal motion of a cursor on a screen with respectively the lateral tilt and the ante/retroversion of their pelvis. We monitored also the degrees of freedom not directly involved in cursor control, thus subjects could complete the task only with a correct posture. Our preliminary results highlight significant improvement not only in cursor control, but also in the Trunk Impairment Scale (TIS) and in the Five Times Sit to Stand Test (5xSST).

A space-dependent atomic superfluid current in Bose–Einstein condensates**Project supported by the National Natural Science Foundation of China (Grant Nos. 11204076 and 11147011), the Scientific Research Fund of Hunan First Normal University (Grant No. XYS13N16), and the Opening Project of Key Laboratory of Low-dimensional Quantum Structures and Quantum Control (Hunan Normal University), Ministry of Education (Grant No. QSQC1005).

A space-dependent atomic superfluid current with an explicit analytical expression and its role in Bose–Einstein condensates are studied. The factors determining the intensity and oscillating amplitude of the space-dependent atomic superfluid current are explored in detail. Research findings reveal that the intensity of the current can be regulated by setting an appropriate configuration of the trap and its oscillating amplitude can be adjusted via Feshbach resonance. It is numerically demonstrated that the space-dependent atomic superfluid current can exert great influence on the spatial distribution of condensed atoms, and even force condensed atoms into very complex distributional states with spatial chaos.

An Adaptive Low-Dimensional Control for a Hybrid Neuroprosthesis

Hybrid neuroprostheses that use both electric motor drives and functional electrical stimulation for the restoration of walking in persons with paraplegia have a promising potential. However, the hybrid actuation structure introduces effector redundancy, making the system complex and difficult to control. In this paper we design a low-dimensional controller inspired from the muscle synergy principle. The new controller requires few control signals to actuate multiple effectors in a hybrid neuroprostheses. The development of the controller and a Lyapunov stability analysis, which yielded semi-global uniformly ultimately boundedness is presented in this paper. Computer simulations were performed to test the new controller on a 2 degree of freedom fixed hip model.

Assessment of optimal control mechanism complexity by experimental landscape Hessian analysis: fragmentation of CH2BrI

Optimally shaped femtosecond laser pulses can often be effectively identified in adaptive feedback quantum control experiments, but elucidating the underlying control mechanism can be a difficult task requiring significant additional analysis. We introduce landscape Hessian analysis (LHA) as a practical experimental tool to aid in elucidating control mechanism insights. This technique is applied to the dissociative ionization of CH2BrI using shaped fs laser pulses for optimization of the absolute yields of ionic fragments as well as their ratios for the competing processes of breaking the C–Br and C–I bonds. The experimental results suggest that these nominally complex problems can be reduced to a low-dimensional control space with insights into the control mechanisms. While the optimal yield for some fragments is dominated by a non-resonant intensity-driven process, the optimal generation of other fragments maa difficult task requiring significant additionaly be explained by a non-resonant process coupled to few level resonant dynamics. Theoretical analysis and modeling is consistent with the experimental observations.

Finite-Horizon Parameterizing Manifolds, and Applications to Suboptimal Control of Nonlinear Parabolic PDEs

This article proposes a new approach based on finite-horizon parameterizing manifolds (PMs) for the design of low-dimensional suboptimal controllers to optimal control problems of nonlinear partial differential equations (PDEs) of parabolic type. Given a finite horizon $[0,T]$ and a low-mode truncation of the PDE, a PM provides an approximate parameterization of the uncontrolled high modes by the controlled low ones so that the unexplained high-mode energy is reduced, in an $L^2$-sense, when this parameterization is applied. Analytic formulas of such PMs are derived by application of the method of pullback approximation of the high-modes (Chekroun, Liu, and Wang, 2013, http://arxiv.org/pdf/1310.3896v1.pdf). These formulas allow for an effective derivation of reduced ODE systems, aimed to model the evolution of the low-mode truncation of the controlled state variable, where the high-mode part is approximated by the PM function applied to the low modes. A priori error estimates between the resulting PM-based low-dimensional suboptimal controller $u_R^\ast$ and the optimal controller $u^*$ are derived. These estimates demonstrate that the closeness of $u_R^\ast$ to $u^*$ is mainly conditioned on two factors: (i) the parameterization defect of a given PM, associated respectively with $u_R^\ast$ and $u^*$; and (ii) the energy kept in the high modes of the PDE solution either driven by $u_R^\ast$ or $u^*$ itself. The practical performances of such PM-based suboptimal controllers are numerically assessed for various optimal control problems associated with a Burgers-type equation. The numerical results show that a PM-based reduced system allows for the design of suboptimal controllers with good performances provided that the associated parameterization defects and energy kept in the high modes are small enough, in agreement with the rigorous results.

A computational analysis of motor synergies by dynamic response decomposition

Analyses of experimental data acquired from humans and other vertebrates have suggested that motor commands may emerge from the combination of a limited set of modules. While many studies have focused on physiological aspects of this modularity, in this paper we propose an investigation of its theoretical foundations. We consider the problem of controlling a planar kinematic chain, and we restrict the admissible actuations to linear combinations of a small set of torque profiles (i.e., motor synergies). This scheme is equivalent to the time-varying synergy model, and it is formalized by means of the dynamic response decomposition (DRD). DRD is a general method to generate open-loop controllers for a dynamical system to solve desired tasks, and it can also be used to synthesize effective motor synergies. We show that a control architecture based on synergies can greatly reduce the dimensionality of the control problem, while keeping a good performance level. Our results suggest that in order to realize an effective and low-dimensional controller, synergies should embed features of both the desired tasks and the system dynamics. These characteristics can be achieved by defining synergies as solutions to a representative set of task instances. The required number of synergies increases with the complexity of the desired tasks. However, a possible strategy to keep the number of synergies low is to construct solutions to complex tasks by concatenating synergy-based actuations associated to simple point-to-point movements, with a limited loss of performance. Ultimately, this work supports the feasibility of controlling a non-linear dynamical systems by linear combinations of basic actuations, and illustrates the fundamental relationship between synergies, desired tasks and system dynamics.

Reconstructing whole-body motions with wrist trajectories

Reconstructing whole-body motions using only a low-dimensional input reduces the cost of and efforts for performance capture significantly, and yet remains a challenging problem. We introduce a novel technique that synthesizes whole-body motion using the two wrist trajectories. Given the wrist trajectories, we first determine the optimal ankle trajectories from a large number of candidate ankle paths obtained from example poses in the motion database. The optimal trajectory is efficiently achieved by solving for the shortest path problem in a directed acyclic graph. Next, we use both the wrist and ankle trajectories as the low-dimensional control signals to achieve the whole-body pose at each time step. We show that our method can reconstruct various whole-body motions that can be recognized by arm motions, such as walking, stepping, and in-place upper-body motions. Comparisons with ground truth motions and with other methods are provided.

Intuitive, Online, Simultaneous, and Proportional Myoelectric Control Over Two Degrees-of-Freedom in Upper Limb Amputees

We propose an approach for online simultaneous and proportional myoelectric control of two degrees-of-freedom (DoF) of the wrist, using surface electromyographic signals. The method is based on the nonnegative matrix factorization (NMF) of the wrist muscle activation to extract low-dimensional control signals translated by the user into kinematic variables. This procedure does not need a training set of signals for which the kinematics is known (labeled dataset) and is thus unsupervised (although it requires an initial calibration without labeled signals). The estimated control signals using NMF are used to directly control two DoFs of wrist. The method was tested on seven subjects with upper limb deficiency and on seven able-bodied subjects. The subjects performed online control of a virtual object with two DoFs to achieve goal-oriented tasks. The performance of the two subject groups, measured as the task completion rate, task completion time, and execution efficiency, was not statistically different. The approach was compared, and demonstrated to be superior to the online control by the industrial state-of-the-art approach. These results show that this new approach, which has several advantages over the previous myoelectric prosthetic control systems, has the potential of providing intuitive and dexterous control of artificial limbs for amputees.

Effects of robotic guidance on the coordination of locomotion

BackgroundFunctional integration of motor activity patterns enables the production of coordinated movements, such as walking. The activation of muscles by weightened summation of activation signals has been demonstrated to represent the spatiotemporal components that determine motor behavior during walking. Exoskeleton robotic devices are now often used in the rehabilitation practice to assist physical therapy of individuals with neurological disorders. These devices are used to promote motor recovery by providing guidance force to the patients. The guidance should in principle lead to a muscle coordination similar to physiological human walking. However, the influence of robotic devices on locomotor patterns needs still to be characterized. The aim of this study was to analyze the effect of force guidance and gait speed on the modular organization of walking in a group of eight healthy subjects.MethodA group of healthy subjects walked on a treadmill with and without robotic aiding at speeds of 1.5, 2.0 and 2.5 Km/h. The guidance force was varied between 20%, 40%, 70% and 100% level of assistance. EMG recordings were obtained from seven leg muscles of the dominant leg and kinematic and kinetic features of the knee and hip joints were extracted.ResultsFour motor modules were sufficient to represent the variety of behavioral goals demanded during robotic guidance, with similar relationships between muscle patterns and biomechanical parameters across subjects, confirming that the low-dimensional and impulsive control of human walking is maintained using robotic force guidance. The conditions of guidance force and speed that maintained correct and incorrect (not natural) modular control were identified.ConclusionIn neurologically intact subjects robotic-guided walking at various force guidance and speed levels does not alter the basic locomotor control and timing. This allows the design of robotic-aided rehabilitation strategies aimed at the modulation of motor modules, which are altered in stroke.

Fractal Fluctuations in Quiet Standing Predict the Use of Mechanical Information for Haptic Perception

Movement science has traditionally understood high-dimensional fluctuations as either antithetical or irrelevant to low-dimensional control. However, fluctuations incident to changeful, sometimes unpredictable stimulation must somehow reshape low-dimensional aspects of control through perception. The movement system's fluctuations may reflect cascade dynamics in which many-sized events interact nonlinearly across many scales. Cascades yield fractal fluctuations, and fractality of fluctuations may provide a window on the interactions across scale supporting perceptual processes. To test these ideas, we asked adult human participants to judge whole or partial length for unseen rods (with and without added masses). The participants' only experience with the objects came from supporting them across their shoulders during quiet standing. First, the degree of fractal temporal correlations in trial-by-trial series of planar Euclidean displacements in center of pressure (COP) significantly improved prediction of subsequent trial-by-trial judgments, above and beyond prediction by traditional predictors of haptic perception and conventional measures of COP variability. Second, comparison with linear surrogate data indicated the presence of nonlinear interactions across scale in these time series. These results demonstrate that high-dimensional fluctuations may serve a crucial role in the cascade dynamics supporting apparently low-dimensional control strategies.

Practical limits on muscle synergy identification by non-negative matrix factorization in systems with mechanical constraints.

Statistical decomposition, including non-negative matrix factorization (NMF), is a convenient tool for identifying patterns of structured variability within behavioral motor programs, but it is unclear how the resolved factors relate to actual neural structures. Factors can be extracted from a uniformly sampled, low-dimension command space. In practical application, the command space is limited, either to those activations that perform some task(s) successfully or to activations induced in response to specific perturbations. NMF was applied to muscle activation patterns synthesized from low dimensional, synergy-like control modules mimicking simple task performance or feedback activation from proprioceptive signals. In the task-constrained paradigm, the accuracy of control module recovery was highly dependent on the sampled volume of control space, such that sampling even 50% of control space produced a substantial degradation in factor accuracy. In the feedback paradigm, NMF was not capable of extracting more than four control modules, even in a mechanical model with seven internal degrees of freedom. Reduced access to the low-dimensional control space imposed by physical constraints may result in substantial distortion of an existing low dimensional controller, such that neither the dimensionality nor the composition of the recovered/extracted factors match the original controller.

Core‐Shell Nanostructures: Modeling, Fabrication, Properties, and Applications

1 School of Materials Science and Engineering, Central South University, Changsha, Hunan 410083, China 2 School of Electronic Science and Applied Physics, Hefei University of Technology, Hefei, Anhui 230009, China 3College of Chemistry and Life Sciences, Zhejiang Normal University, Jinhua, Zhejiang 321004, China 4Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of the Ministry of Education, Department of Physics, Hunan Normal University, Hunan Changsha 410081, China 5Materials Research Centre, Indian Institute of Science, Bangalore 560012, India 6Department of Chemistry, Western Michigan University, Kalamazoo, MI 49008, USA

A framework for real-time implementation of low-dimensional parameterized NMPC

In this paper, a novel approach is proposed to implement low-dimensional parameterized Nonlinear Model Predictive Control (NMPC) schemes for systems showing fast dynamics. The proposed scheme is based on distributing the reconstruction of the cost function over the real lifetime of the controlled system. The framework is particularly suitable for NMPC formulations that use low dimensional control parametrization. The concrete example of a Planar Vertical Take-Off and Landing (PVTOL) aircraft stabilization problem is used to illustrate the efficiency of the proposed formulation.